Occasion: 13th International Symposium on Orthogonal Polynomials, Special Functions and Applications, Minisymposium on Symbolic Computation and Special Functions
It was shown by Gessel that the Apéry numbers, introduced by Apéry in his proof of the irrationality of \(\zeta(3)\), are periodic modulo \(8\). We investigate this, and other divisibility results, for Apéry-like numbers. For instance, we prove that the Almkvist-Zudilin numbers are periodic modulo \(8\) as well. The ingredients of our proof are a multivariate rational function whose diagonals are the Almkvist-Zudilin numbers and a theorem originating with Furstenberg which states that, modulo a fixed prime power, these values have algebraic generating function and, hence, can be generated by a finite state automaton. As demonstrated recently by Rowland and Yassawi, these automata can be computed mechanically.
|2015aperylike-siam-opsfa.pdf||1.25 MB||Slides (PDF, 56 pages)||447|